How much extra information can we conclude from a neural network output values?
From the values you have provided I can easily guess your output layer has the sigmoid (do clarify!) activation function. For sigmoid activation function this can be a quite normal occurrence. Also maybe the number of training epochs is not high enough.

The case you have mentioned of 0.99 is generally in the case of the output being subjected to a softmax probability function. Although 0.99 is still achievable using sigmoid activation, it will depend on your network hyper-parameters and in general the data. If data is very easily separable the sigmoid will generally give very high contrasting difference among classes. Also if the hidden network has ReLu it becomes easy to provide contrasting class scores due to its huge scale difference compared to a sigmoid activation in the hidden layer.

The point here to note is contrasting difference among classes, because your sigmoid might give 0.99 for the correct class while 0.9 for other classes, which is undesirable.